Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1s (2025) 

 

552 https://internationalpubls.com 

A Connection Between Knot Theory and Linear Algebra 
 

Ayman Abouzaid Ezelarab Aboufattoum 

Department of Mathematics, Faculty of Science, Kuwait University, P. O. Box 5969, Safat - 13060 State of Kuwait. 

E-mail: ayman.aboufattoum@ku.edu.kw 

 

Article History: 

Received: 29-08-2024 

Revised: 15-10-2024 

Accepted: 01-11-2024 

Abstract:  

In knot theory literature, a lot of references treated the topic of computing the Alexander 

polynomial and proved that the Alexander polynomial is a knot invariant. In this paper, 

we concentrate on an unconventional connection between knot theory and linear 

algebra in computing the Alexander polynomial and introduce an approach depending 

linear algebra tools to show that the Alexander polynomial is a knot invariant. Also, we 

connect this approach to the way of Fox coloring to suggest another way to prove that 

the 3-coloring is a knot invariant and confirm what is already known about the 3-

coloring of the trefoil and the figure eight knots.. 

Keywords: Knot Theory, Linear Algebra. 

 

1. Introduction 

Alexander polynomial is the first polynomial knot invariant  discovered. It is a well understood 

classical knot invariant discovered by J. W. Alexander in 1927. There are many ways to compute the 

Alexander polynomial,  involving algebraic techniques and others have more  geometric approaches 

such as [1], [2], [4], [6],... In this paper, we will see connections between knot theory and linear algebra 

regarding two knot invariants, the Alexander polynomial and the 3-coloring. To prove that the 

Alexander polynomial is a knot invariant is not a new topic, it is already proved in several references 

such as [1], [2], [3], [4], [7]. We claim that we touch a different approach to prove such a well-known 

property of the Alexander polynomial and suggest a similar proof for the same property of the 3-

coloring. Also, we treated the 3-coloring invariant from the linear algebra point of view depending on 

the Fox coloring. 

We organized this paper as follows: In section 2, we give a linear algebra approach of computing the 

Alexander polynomial of a knot and apply it on the trefoil knot. In section 3, we prepare the reader to 

accept that playing in the given oriented diagram of a knot in a specific way may change the form of 

the corresponding homogeneous system of equations that can be obtained from a knot diagram but 

does not change its space of solutions and then the Alexander polynomial of the given knot is 

unchanged. In section 4, we put an approach to prove that the Alexander polynomial is a knot invariant. 

In section 5, we connect the previous approach to the Fox coloring way to suggest a proof of the 

assumption that the 3-coloring is a knot invariant and give a way to check if a given knot diagram is 

3-coloring or not. Then we apply this approach on the trefoil and the figure eight knots depending on 

the system obtained in section 2. 

2. The Alexander polynomial 

Given an oriented diagram of a knot 𝐾 of 𝑛 crossings. We assign the variables: 𝑥1, 𝑥2, … , 𝑥𝑛 to its arcs. 

At any crossing, we see three arcs in the position as in Figure 2.1. 

mailto:ayman.aboufattoum@ku.edu.kw


Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1s (2025) 

 

553 https://internationalpubls.com 

 

For such  crossing, we write the equation : 𝑡𝑥𝑖 + (1− 𝑡)𝑥𝑖 = 𝑥𝑘 , 𝑡 ∈ ℂ , then we construct the 

following homogeneous system of 𝑛 linear equations in the variables: 𝑥1, 𝑥2, … , 𝑥𝑛 . 

         𝑡𝑥𝑖 + (1− 𝑡)𝑥𝑖 = 𝑥𝑘 , 𝑖, 𝑘 = 1, 𝑛 , 𝑡 ∈ ℂ                              (2.1) 

This system can be also written in the matrix form, 𝐴𝑋 = 0⃑ . Because of this way of construction, the 

entries of any row of 𝐴 are  𝑡, 1− 𝑡,−1, and some possible zeros then 𝐴 is singular for every 𝑡. In the 

1920s, Alexander showed that all (𝑛 − 1)-minors are equal up to multiplication by the monomial  

±𝑡𝑘 , 𝑘 ∈ ℤ. So, these minors which are Laurent polynomials in 𝑡 can be normalized for some 

consideration by dividing a minor by ±𝑡𝑘. Any one of these polynomials is called an Alexander 

polynomial of the Knot 𝐾, denoted by  ∇𝐾(𝑡) with ∇Ο(𝑡) = 1. One often fixes a particular form as the 

first Alexander's choice of the normalization so that the term of the polynomial of the least degree of 𝑡 

is a positive constant. 

    A simple example is now introduced to visualize the process of computing the Alexander polynomial 

using this approach. let us use this approach to compute the Alexander polynomial of the trefoil. 

 

The corresponding linear system is 

𝑡𝑧 + (1− 𝑡)𝑥 = 𝑧 

𝑡𝑥 + (1− 𝑡)𝑦 = 𝑧 

𝑡𝑦 + (1− 𝑡)𝑧 = 𝑥 

Which can be rewritten as: 

[
1− 𝑡 −1 𝑡

𝑡 1− 𝑡 −1
−1 𝑡 1− 𝑡

] [
𝑥
𝑦
𝑧
] = [

0

0

0

] 

Take any 2−minor, say |
𝑡 −1

−1 1− 𝑡
| = 𝑡 − 𝑡2 + 1, or alternatively, if we take |

𝑡 −1
−1 1− 𝑡

| = 𝑡 − 𝑡2 −

1, which can be normalized to the previous one. 

3.  A useful definition 

We claim that getting a homogeneous linear system of a knot and preserving its solutions from the 

change guarantees that the Alexander polynomial of the same knot will not change up to multiplication 

of ±𝑡𝑘, for some integer 𝑘. A very close  claim is mentioned and proved in [2]. As in the basics of 

linear algebra, where we can check if two given matrices of the same size are row equivalent or not, 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

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the following definition from [7] helps us to extend the concept of equivalency regarding the solutions 

of the corresponding homogeneous linear systems of equations. 

Definition. Two matrices are equivalent if one can be obtained from the other by a finite sequence of 

the following operations and their inverse operations: 

1. Permute two rows(columns). 

2. Adjoin a new row of zeros. 

3. Add a multiple of a row(column) to any row(column). 

4. Multiply a row(column) by a unit. 

5. Replace the 𝑚 × 𝑛 matrix 𝐴 by the extended (𝑚 + 1) × (𝑛 + 1)-matrix: 

[

𝐴 0

⋮

0 …
0

0 1

] 

4.  A linear algebra approach to the proof of an old theorem 

Theorem.  ∇𝐾(𝑡) is a knot invariant. 

Proof. To show that  ∇𝐾(𝑡) is a knot invariant it is sufficient to show that it does not change under 

Reidemeister moves. For the first Reidemeister move, a twist is introduced to any arc of the given 

oriented knot diagram, say, 𝑥𝑖  as in Figure 4.1.  

 

Keeping the whole diagram fixed except this move at the arc 𝑥𝑖. To see the contribution of this move 

to the system 2.1 of the given diagram, the new twist adds the equation: (1− 𝑡)𝑥𝑖 + 𝑡𝑥𝑖 = 𝑥𝑖 , which 

does not affect the solutions of the system. It is typically, as one adjoined a row of zeros to the 

coefficients matrix of the system  𝐴𝑋 = 0⃑ .  Then ∇𝐾(𝑡)  is unchanged under the first Reidemeister 

move. Note. The other type of the twist will add the same equation to the system 2.1.  For the second 

Reidemeister move, two new crossings are introduced to any two arcs say, 𝑥𝑖 and 𝑥𝑘 as in Figure 4.2. 

 

Again, keeping the diagram fixed except this move at the arcs 𝑥𝑗 and 𝑥𝑘. The contribution of this move 

to the system 2.1  is given by adding two equations of the form: (1− 𝑡)𝑥𝑖 + 𝑡𝑥𝑖 = 𝑥𝑖. Each one of 

these two equations can be rewritten in the form: (1− 𝑡)𝑥𝑖 + 𝑡𝑥𝑖 = 0, which do not affect the solution 

of the system 2.1 as we mentioned in Section 3. So,  ∇𝐾(𝑡) is unchanged under the second Reidemeister 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

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move. Note. The other position of the second Reidemeister move will add the same equations to the 

system 2.1.  For the third Reidemeister move, the contribution of the three crossings before and after 

the move are the same, so ∇𝐾(𝑡) is unchanged under the third Reidemeister move.    Finally, we point 

out that we used the first and the second Reidemeister moves as introducing a twist and two new 

crossings, respectively, not removing them. Now we will treat the case of removing. Since we have 

proved that after introducing a twist or two crossings, the Alexander polynomial ∇𝐾(𝑡) does not change 

then if we take the last diagram after introducing a twist or two new crossings as the original stage and 

we remove a twist or two crossings by the first or the second Reidemeister move, respectively, we 

return to the main diagram which has the same Alexander polynomial, ∇𝐾(𝑡). 

5. A connection with Fox coloring 

In [6], Fox defined tricolorability or 3-coloring of a knot diagram as a coloring of one of its diagrams 

with three colors such that: (i) At least two different colors are used. (ii) At each crossing, the three 

arcs are of the same color or all are of distinct colors. 

Note. This definition does not require the given diagram to be oriented. Furthermore, it is proved that 

3-coloring is a knot invariant. It seems that 3-coloring is not numerical knot invariant, but this view 

may change when we connect 3-coloring with the system of homogeneous linear equations 2.1, where 

3-coloring becomes a linear algebra problem ignoring the orientation which is necessary for the system 

2.1. 

Recall one equation from the system 2.1 say, 𝑡𝑥𝑖 + (1− 𝑡)𝑥𝑖 = 𝑥𝑘. If we put  𝑡 = −1 we get  2𝑥𝑖 −
𝑥𝑖 − 𝑥𝑘 = 0. This equation is called Fox coloring condition of the crossing with an upper arc 𝑥𝑗 and 

two lower arcs 𝑥𝑖  and  𝑥𝑘. So, we can assign any given knot diagram with a new system of linear 

homogeneous equations over ℤ called the coloring system corresponds to the system 2.1 with 𝑡 = −1, 

which is always consistent. If we consider this new system of homogeneous linear equations over ℤ3, 

the 3-coloring problem becomes the problem of the existence or non-existence of nontrivial solutions 

of the system 𝐴𝑋 = 0⃑ , where 𝐴 is any 3 × 3-matrix obtained by omitting some rows and the same 

number of columns from the coefficients matrix of the coloring system. 

Now why not? We can collect what we did in the proof of the main theorem in section 4 with what we 

mentioned about the 3-coloring property that it depends on the solutions of a system obtained from the 

system 2.1 of the given knot diagram to state that Reidemeister moves do not change the solutions 

space of the system 2.1 and then 3-coloring of the given diagram. This can be considered as a proof of 

a similar theorem to the theorem in section 4 that is 3-coloring is a knot invariant. 

  Applications. 

5.1 The trefoil: We can return to the obtained system for the trefoil in section 2 to solve the 3-coloring 

system that is 

[
2 −1 −1

−1 2 −1
−1 −1 2

] [
𝑥
𝑦
𝑧
] = [

0

0

0

] 

Since the system has non-trivial solutions then the trefoil is 3-coloring. 

5.2 The figure eight: Given any oriented diagram of the figure eight, we can easily  

check that its coloring system is 



Communications on Applied Nonlinear Analysis 

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[

2 −1 0 −1
0 2 −1 −1

−1
−1

−1
0

2

−1
0

2

]  [

𝑥
𝑦
𝑧
𝑤

] = [

0
0

0

0

] 

Since any 3 × 3-matrix  𝐴 obtained from the coefficient matrix of the last system by omitting one row 

and one column is non-singular then the system 𝐴𝑋 = 0⃑   has only the trivial solution for each 𝐴. This 

confirms that the figure eight is not 3-coloring. 

References 

[1]   C. Livingston, Knot Theory, The Mathematical Association of America, Washington D. C, 1993. 

[2]   E. Long, Three Routes to Alexander Polynomial, Manchester University, 2005. 

[3]   J. May, Matrix Representation of Knot and Link Groups, Harvey Mudd College, 2006. 

[4]   J. W. Alexander, Topological Invariants of Knots and Links, Transactions of American Mathematical Society, 30(2), 

1928, 275-306. 

[5]   L. H. Kauffman, P. Lopes, Coloring beyond Fox: the other linear Alexander quandles, arXiv: 1708. 01932v3[math. 

GT], April 10, 2018. 

[6]   R. H. Crowell and R. H. Fox, Introduction to Knot Theory, Ginn and company, 1963. 

[7]   R. Fox, A Quick Trip Through Knot Theory, Topology of 3-Manifolds and Related Topics, 86, 1949, 120-1